In this post, we will see the book *Mathematical Foundations of Statistical Mechanic*s by A. I. Khinchin.

# About the book

The present book considers as its main task to make the reader familiar with the mathematical treatment of statistical mechanics on the basis of modern concepts of the theory of probability and a maximum utilization of its analytic apparatus. The book is written, above all, for the mathematician, and its purpose is to introduce him to the problems of statistical mechanics in an atmosphere of logical precision, outside of which he cannot assimilate and work, and which, unfortunately, is lacking in the existing physical expositions.The only essentially new material in this book consists in the systematic use of limit theorems of the theory of probability for rigorous proofs of asymptotic formulas without any special analytic apparatus. The few existing expositions which intended to give a rigorous proof to these formulas, were forced to use for this purpose special, rather cumbersome, mathematical machinery. We hope, however, that our exposition of several other questions (the ergodic problem, properties of entropy, intramolecular correlation, etc.) can claim to be new to a certain extent, at least in some of its parts.

The book was translated from Russian by George Gamow was first published in 1949.

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You can get the book here.

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# Contents

Preface vii

## Chapter I. Introduction

1. A brief historical sketch 1

2. Methodological characterization 7

## Chapter II. Geometry and Kinematics of the Phase Space

3. The phase space of a mechanical system 13

4. Theorem of Liouville 15

5. Theorem of Birkhoff 19

6. Case of metric indecomposability 28

7. Structure functions 32

8. Components of mechanical systems 38

## Chapter III. Ergodic Problem

9. Interpretation of physical quantities in statistical mechanics 44

10. Fixed and free integrals 47

11. Brief historical sketch 52

12. On metric indecomposability of reduced manifolds 55

13. The possibility of a formulation without the use of metric indecomposability 62

## Chapter IV. Reduction to the Problem of the Theory of

Probability

14. Fundamental distribution law 70

15. The distribution law of a component and its energy 71

16. Generating functions 76

17. Conjugate distribution functions 79

18. Systems consisting of a large number of components 81

## Chapter V. Application of the Central Limit Theorem

19. Approximate expressions of structure functions 84

20. The small component and its energy. Boltzmann’s law 88

21. Mean values of the sum functions 93

22. Energy distribution law of the large component 99

23. Example of monatomic ideal gas 100

24. The theorem of equipartition of energy 104

25. A system in thermal equilibrium. Canonical distribution of Gibbs 110

## Chapter VI. Ideal Monatomic Gas

26. Velocity distribution. Maxwell’s law 115

27. The gas pressure 116

28. Physical interpretation of the parameter 121

29. Gas pressure in an arbitrary field of force 123

## Chapter VII. The Foundation of Thermodynamics

30. External parameters and the mean values of external forces 129

31. The volume of the gas as an external parameter 131

32. The second law of thermodynamics 132

33. The properties of entropy 137

34. Other thermodynamical functions 145

## Chapter VIII. Dispersion and the Distributions of Sum Functions

35. The inter molecular correlation 148

36. Dispersion and distribution laws of the sum functions 156

Appendix

The proof of the central limit theorem of the theory of probability 166

Notations 176

Index 178

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